Comparing Category Theory and Model Theory for Master's Thesis.

Question

I am currently doing a Masters thesis in pure maths, and the two current fields that excite me are Category Theory (CT) and Model Theory (MT).

I have been reading up on David Marker's Model Theory: An Introduction, and Saunder MacLane's Categories for the Working Mathematician, so I do have a basic understanding of both. So far, I love the notion that they both provide an underlying framework for most fields in pure maths.

However, I have googled but there are not much resources out there about open questions of the application of CT and MT.

Therefore,

QUESTION ONE:- What are the applications of CT and MT, and could someone please give me a rough idea of the future landscape of CT and MT?

Secondly, my lecturers are encouraging me to do CT and MT with applications in mind. However, I am very interested in studying PURE CT and MT.

So,

QUESTION TWO:- Are there any open questions within CT and MT themselves that are worth exploring? (Both at a masters thesis level and a doctorates?)

Thank you for your time guys!

Answer 1

The interplay between CT and MT is pretty well established. The term to search for is locally accessible categories. Another subject to look at may be topos theory, again with plenty of material online. A complete, or even a very partial list of applications of CT and MT will require a lot of bytes. MT has applications in algebra and in analysis, and that alone is quite a lot. Chang & Keisler's book has a list of open problems. CT as a language is useful pretty much throughout mathematics, but you are likely looking for deeper applications. It may be debatable what constitutes a deep application of CT (there used to be lots of meta-discussions on just how much nonsense is general abstract nonsense), but without a doubt CT is of great importance in algebraic topology, with current research on $\infty $-categories (see e.g., Lurie's work).

As to open problems in CT and MT purely, it is again perhaps somewhat unclear what that means precisely, but there are good/excellent journals dedicated to MT and CT, and that would be a place to look at to get a feel for what people are doing.

I hope this answers your question.

Answer 2

For a good concrete example, I would suggest that you look into synthetic differential geometry. This is an axiomatic approach to differential geometry which takes place in a smooth topos. The theory is very beautiful and intuitive, and allows you to rigorously reason using infinitesimals.

Since this is a purely axiomatic theory, you can come up with a variety of different models which satisfy the axioms. The ones differential geometers would be most interested in are so-called well-adapted models, where the category of smooth manifolds embeds fully and faithfully. But there are other models which would be of more interest to algebraic geometers, allowing you to use differential-geometric reasoning and tools in algebraic geometry.

If you are interested in learning more, there is a freely available text by Anders Kock, who is one of the pioneers of this theory.